A computational method based on Bézier control points is presented to solve optimal control problems governed by time varying linear dynamical systems subject to terminal state equality constraints and state inequality constraints. The method approximates each of the system state variables and each of the control variables by a Bézier curve of unknown control points.The new approximated problems converted to a quadratic programming problem which can be solved more easily than the original problem. Some examples are given to verify the efficiency and reliability of the proposed method.Mathematical subject classification: 49N10.

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We will apply the successive approximation method for proving the Hyers-Ulam stability of a nonlinear integral equation.

Abstract. In this paper, linear quadratic optimal control problems are solved by applying least square method based on Bézier control points. We divide the interval which includes t, into k subintervals and approximate the trajectory and control functions by Bézier curves. We have chosen the Bézier curves as piacewise polynomials of degree three, and determined Bézier curves on any subinterval by four control points. By using least square method, we introduce an optimization problem and compute the control points by solving this optimization problem.Numerical experiments are presented to illustrate the proposed method.Mathematical subject classification: 49N10.

Fuzzy fractional differential equations (FFDEs) driven by Liu's process are a type of fractional differential equations. In this paper, we intend to provide and prove a novel existence and uniqueness theorem for the solutions of FFDEs under local Lipschitz and linear growth conditions. We also investigate the stability of solutions to FFDEs by a theorem. Finally, some examples are provided.

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Abstract. The original parametric iteration method (PIM) provides the solution of a nonlinearsecond order boundary value problem (BVP) as a sequence of iterations. Since the successive iterations of the PIM may be very complex so that the resulting integrals in its iterative relation may not be performed analytically. Also, the implementation of the PIM generally leads to calculation of unneeded terms, which more time is consumed in repeated calculations for series solutions. In order to overcome these difficulties, in this paper, a useful improvement of the PIM is proposed.The implementation of the modified method is demonstrated by solving several nonlinear second order BVPs. The results reveal that the new developed method is a promising analytical tool to solve the nonlinear second order BVPs and more promising because it can further be applied easily to solve nonlinear higher order BVPs with highly accurate.Mathematical subject classification: Primary: 34B15; Secondary: 41A10.

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